We use the p-adic local Langlands correspondence for GL2(Qp) to explicitly compute the reduction modulo p of certain 2-dimensional crystalline representations of small slope, and give applications to modular forms.

Twenty-five years ago R. Langlands proposed [L] a “fantastic generalization” of Artin-Hasse reciprocity law in the classical class field theory. He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive groupG over a global number field on the one hand, and (roughly speaking) finite dimensionsional representations of the...

0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq. Denote by F the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `...

The purpose of this article is to state a strong form of the conjecture of Langlands and Rapoport (1987, §5), and to outline a proof of it. For Shimura varieties of PEL type, the outline is complete, and for Shimura varieties of Hodge type it becomes complete if one accepts certain statements, stated as conjectures in the original version of this article, whose proof has been announced by Vasiu...

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

We explain (following V. Drinfeld) how the G(C[[t]]) equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restriction to the Kostant slice. We extend this ...

We provide a motivated introduction to the theory of categorical actions groups and local geometric Langlands program. Along way, we emphasize applications, old new, usual representation reductive affine Lie algebras.